We study the $N$-dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.

We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual $\gamma$-law pressure, $\gamma \ge 1$, admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the $2\times 2$$p$-system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann...

In this paper we prove the global well-posedness of the two-dimensional Boussinesq system with zero viscosity for rough initial data.

We give explicit formulas for Hadamard's coefficients in terms of the tau-function of the Korteweg-de Vries hierarchy. We show that some of the basic properties of these coefficients can be easily derived from these formulas.

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $${\widehat{H}}_s^r\left(ℝ\right)$$ defined by the norm $${∥v_0∥}_{{\widehat{H}}_s^r\left(ℝ\right)}:={∥{〈\xi 〉}^s\widehat{v_0}∥}_{L_{\xi }^{r^{\text{'}}}},〈\xi 〉={\left(1+{\xi }^2\right)}^{\frac{1}{2}},\frac{1}{r}+\frac{1}{r^{\text{'}}}=1$$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $$\frac{2j-1}{2r^{\text{'}}}$$ . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < $$\frac{2j-1}{2r^{\text{'}}}$$ . The results for r = 2 - so far in...

We analyse the effect of the mechanical response of the solid phase during liquid/solid phase change by numerical simulation of a benchmark test based on the well-known and debated experiment of melting of a pure gallium slab counducted by Gau & Viskanta in 1986. The adopted mathematical model includes the description of the melt flow and of the solid phase deformations. Surprisingly the conclusion reached is that, even in this case of pure material, the contribution of the solid phase to the...

We demonstrate a theorem of existence and uniqueness on a large scale of the solution of a system of differential disequations associated to a Graffi model relative to the motion of two incompressible viscous fluids.

In this paper, we propose a computational model to investigate the coupling between cell’s adhesions and actin fibres and how this coupling affects cell shape and stability. To accomplish that, we take into account the successive stages of adhesion maturation from adhesion precursors to focal complexes and ultimately to focal adhesions, as well as the actin fibres evolution from growing filaments, to bundles and finally contractile stress fibres.We use substrates with discrete patterns of adhesive...

We consider higher order mixed finite element methods for the incompressible Stokes or Navier-Stokes equations with Qr-elements for the velocity and discontinuous $P_{r-1}$-elements for the pressure where the order r can vary from element to element between 2 and a fixed bound $r^{*}$. We prove the inf-sup condition uniformly with respect to the meshwidth h on general quadrilateral and hexahedral meshes with hanging nodes.

By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential $$\mathrm{i}{\varphi }_t=-▵\varphi +{\left|x\right|}^2\varphi -{\left|x\right|}^b{\left|\varphi \right|}^{p-2}\varphi .$$ We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.

We consider the spatial behavior of the velocity field $u(x,t)$ of a fluid filling the whole space ${ℝ}^n$ ($n\ge 2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\mathrm{d}x=c\left(t\right){\delta }_{h,k}$ under more general assumptions on the localization of $u$. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.

We consider the spatial behavior of the velocity field u(x, t) of a fluid filling the whole space ${}^n$ ($n\ge 2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\mathrm{d}x=c\left(t\right){\delta }_{h,k}$ under more general assumptions on the localization of u. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.